1852 CHAPTER 58. ELLIPTIC FUNCTIONS

Therefore, there exists a constant, C j such that

σ (z+w j) =C jσ (z)eη jz.

Next show σ is an odd function, (σ (−z) =−σ (z)) and then let z = −w j/2 to find

C j =−eη jw j

2 and so

σ (z+w j) =−σ (z)eη j

(z+

w j2

). (58.3.41)

7. Show any even elliptic function, f with periods w1 and w2 for which 0 is neither apole nor a zero can be expressed in the form

f (0)n

∏k=1

℘(z)−℘(ak)

℘(z)−℘(bk)

where C is some constant. Here ℘ is the Weierstrass function which comes from thetwo periods, w1 and w2. Hint: You might consider the above function in terms of thepoles and zeros on a period parallelogram and recall that an entire function which iselliptic is a constant.

8. Suppose f is any elliptic function with {w1,w2} a basis for the module of periods.Using Theorem 58.1.8 and 58.3.41 show that there exists constants a1, · · · ,an andb1, · · · ,bn such that for some constant C,

f (z) =Cn

∏k=1

σ (z−ak)

σ (z−bk).

Hint: You might try something like this: By Theorem 58.1.8, it follows that if {αk}are the zeros and {bk} the poles in an appropriate period parallelogram, ∑αk−∑bkequals a period. Replace αk with ak such that ∑ak−∑bk = 0. Then use 58.3.41 toshow that the given formula for f is bi periodic. Anyway, you try to arrange thingssuch that the given formula has the same poles as f . Remember an entire ellipticfunction equals a constant.

9. Show that the map τ → 1− 1τ

maps l2 onto the curve, C in the above picture on themapping properties of λ .

10. Modify the proof of Theorem 58.1.22 to show that λ (Ω)∩{z ∈ C : Im(z)< 0}= /0.